problem stringlengths 11 5.15k | answer stringlengths 1 42 | solution stringlengths 0 8.75k |
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While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $\$14$ each, $4$-pound rocks worth $\$11$ each, and $1$-pound rocks worth $\$2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave? | 50 | To determine the maximum value of rocks Carl can carry, we first calculate the value per pound for each type of rock:
- **5-pound rocks:** $\frac{\$14}{5} = \$2.80$ per pound
- **4-pound rocks:** $\frac{\$11}{4} = \$2.75$ per pound
- **1-pound rocks:** $\frac{\$2}{1} = \$2.00$ per pound
Given that Carl can carry at mo... |
Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms? | 64 | We are given that each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. We need to find the total number of students and rabbits in all $4$ classrooms and then determine how many more students there are than rabbits.
#### Step 1: Calculate the total number of students in all class... |
For real numbers $a$ and $b$, define $a * b=(a-b)^2$. What is $(x-y)^2*(y-x)^2$? | 0 | 1. **Understanding the operation**: Given the operation $a * b = (a-b)^2$, we need to evaluate $(x-y)^2 * (y-x)^2$.
2. **Symmetry in Squaring**: Note that squaring a real number results in the same value whether the number is positive or negative. Therefore, $(-a)^2 = a^2$. Applying this to our problem, we have:
\[... |
If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then the sum of the first $4n$ positive integers is | 300 | 1. **Set up the equation based on the problem statement:**
The sum of the first $3n$ positive integers is given by the formula $\frac{3n(3n+1)}{2}$, and the sum of the first $n$ positive integers is given by $\frac{n(n+1)}{2}$. According to the problem, the sum of the first $3n$ integers is $150$ more than the sum o... |
Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. ... | 4 | 1. **Calculate the total cost of the pizza:**
The pizza is plain and costs $8$ dollars. Adding anchovies to half of it costs an additional $2$ dollars. Therefore, the total cost of the pizza is:
\[
8 + 2 = 10 \text{ dollars}
\]
2. **Determine the cost per slice:**
The pizza has $8$ slices, and the t... |
The five solutions to the equation\[(z-1)(z^2+2z+4)(z^2+4z+6)=0\] may be written in the form $x_k+y_ki$ for $1\le k\le 5,$ where $x_k$ and $y_k$ are real. Let $\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$. The eccentricity of $\mathcal E$ ca... | 7 | 1. **Factorize the given equation**:
\[(z-1)(z^2+2z+4)(z^2+4z+6)=0.\]
We complete the square for the quadratic terms:
\[(z-1)\left((z+1)^2+3\right)\left((z+2)^2+2\right)=0.\]
2. **Find the roots**:
- From $(z-1)=0$, we get $z=1$.
- From $(z+1)^2+3=0$, we get $z=-1\pm\sqrt{3}i$.
- From $(z+2)^2+2=0$, ... |
A small bottle of shampoo can hold $35$ milliliters of shampoo, whereas a large bottle can hold $500$ milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy? | 15 | To find the minimum number of small bottles needed to completely fill a large bottle, we need to divide the capacity of the large bottle by the capacity of the small bottle and then round up to the nearest whole number if necessary.
1. **Calculate the number of small bottles needed:**
\[
\text{Number of small bo... |
What is the smallest integral value of $k$ such that
$2x(kx-4)-x^2+6=0$ has no real roots? | 2 | 1. **Expand and simplify the given quadratic equation**:
\[
2x(kx-4) - x^2 + 6 = 0
\]
Expanding the terms:
\[
2kx^2 - 8x - x^2 + 6 = 0
\]
Combine like terms:
\[
(2k-1)x^2 - 8x + 6 = 0
\]
2. **Condition for no real roots**:
A quadratic equation $ax^2 + bx + c = 0$ has no real roots i... |
Given that $3^8 \cdot 5^2 = a^b,$ where both $a$ and $b$ are positive integers, find the smallest possible value for $a+b$. | 407 | 1. **Express the given equation in terms of $a$ and $b$:**
Given the equation $3^8 \cdot 5^2 = a^b$, we need to find $a$ and $b$ such that $a^b$ equals $3^8 \cdot 5^2$.
2. **Simplify the expression:**
\[
3^8 \cdot 5^2 = (3^4)^2 \cdot 5^2 = (81)^2 \cdot 25 = 81 \cdot 81 \cdot 25
\]
We can further simplif... |
If $2137^{753}$ is multiplied out, the units' digit in the final product is: | 7 | 1. **Identify the units digit of the base number**: The units digit of $2137$ is $7$. Therefore, the units digit of $2137^{753}$ will be the same as the units digit of $7^{753}$.
2. **Determine the pattern of units digits for powers of $7$**:
- $7^1$ has a units digit of $7$.
- $7^2 = 49$, which has a units digi... |
The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? | 7 | 1. **Identify the relationship between the angles**: In a right triangle, the sum of the angles is $180^\circ$. Given one angle is $90^\circ$, the sum of the other two angles, $a^\circ$ and $b^\circ$, must be $90^\circ$. Thus, we have:
\[
a + b = 90^\circ
\]
2. **Condition on $a$ and $b$**: Both $a$ and $b$ a... |
Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the... | 70 | 1. **Define Variables:**
Let $L$ be the length of the ship, $E$ be the length of Emily's step, and $S$ be the length of the ship's step (the distance the ship travels while Emily takes one step).
2. **Set Up Equations:**
- When Emily walks from the back of the ship to the front, she takes $210$ steps, covering a... |
Medians $BD$ and $CE$ of triangle $ABC$ are perpendicular, $BD=8$, and $CE=12$. The area of triangle $ABC$ is | 64 | Given that medians $BD$ and $CE$ of triangle $ABC$ are perpendicular, and their lengths are $BD = 8$ and $CE = 12$, we need to find the area of triangle $ABC$.
1. **Understanding the properties of medians**:
- The medians of a triangle intersect at the centroid, which divides each median into two segments, one of ... |
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ | 117 | 1. **Identify the prime factorization of 100,000**:
The prime factorization of $100,000$ is $100,000 = 10^5 = (2 \cdot 5)^5 = 2^5 \cdot 5^5$.
2. **Determine the set $S$ of divisors**:
The set $S$ consists of all divisors of $100,000$, which can be expressed in the form $2^a5^b$ where $0 \leq a, b \leq 5$.
3. *... |
You and five friends need to raise $1500$ dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise? | 250 | 1. **Identify the total amount to be raised and the number of people involved:**
- Total amount to be raised: $1500$ dollars.
- Number of people involved: $6$ (you and five friends).
2. **Calculate the amount each person needs to raise:**
- To find out how much each person needs to raise, divide the total amo... |
What is $\left(20 - \left(2010 - 201\right)\right) + \left(2010 - \left(201 - 20\right)\right)$? | 40 | We start by simplifying the expression inside the parentheses:
1. Calculate $2010 - 201$:
\[
2010 - 201 = 1809
\]
2. Substitute this result back into the first part of the expression:
\[
20 - (2010 - 201) = 20 - 1809 = -1789
\]
3. Calculate $201 - 20$:
\[
201 - 20 = 181
\]
4. Substitute t... |
There are $10$ horses, named Horse $1$, Horse $2$, . . . , Horse $10$. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. The horses start running... | 6 | To solve this problem, we need to find the least time $T > 0$ such that at least $5$ of the horses are again at the starting point. Each horse $k$ returns to the starting point at multiples of $k$ minutes. Therefore, we are looking for the smallest time $T$ that is a common multiple of the running times of any $5$ hors... |
In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. The perimeter of $ABCD$ is | 34 | 1. **Identify the Shape and Given Information**: We are given a trapezoid $ABCD$ where $AB$ and $CD$ are equal in length. We are also given that the perimeter of the trapezoid is one of the options provided.
2. **Assumption of Rectangle Presence**: The solution assumes the presence of a rectangle within the trapezoid,... |
On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz? | 3 | 1. **Calculate the percentage of students scoring 100 points**:
Given that $10\%$ scored $70$ points, $35\%$ scored $80$ points, and $30\%$ scored $90$ points, the percentage of students scoring $100$ points is:
\[
100\% - (10\% + 35\% + 30\%) = 100\% - 75\% = 25\%
\]
2. **Determine the median score**:
... |
A dress originally priced at $80$ dollars was put on sale for $25\%$ off. If $10\%$ tax was added to the sale price, then the total selling price (in dollars) of the dress was | 54 | 1. **Calculate the Sale Price After Discount:**
The original price of the dress is $80$ dollars. The discount offered is $25\%$. The amount of discount in dollars can be calculated as:
\[
\text{Discount} = 25\% \times 80 = 0.25 \times 80 = 20 \text{ dollars}
\]
Therefore, the sale price after applying th... |
Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in fee... | 8 | 1. **Calculate the number of gaps between the poles**:
There are 41 poles, so there are \(41 - 1 = 40\) gaps between the poles.
2. **Calculate the total number of strides and leaps**:
- Elmer takes 44 strides per gap, so for 40 gaps, he takes \(44 \times 40 = 1760\) strides.
- Oscar takes 12 leaps per gap, s... |
Last summer $100$ students attended basketball camp. Of those attending, $52$ were boys and $48$ were girls. Also, $40$ students were from Jonas Middle School and $60$ were from Clay Middle School. Twenty of the girls were from Jonas Middle School. How many of the boys were from Clay Middle School? | 32 | 1. **Organize the given data into a table**: We start by setting up a table to organize the information provided in the problem. We know the total number of students, the breakdown by gender, the breakdown by school, and the number of girls from Jonas Middle School.
\[
\begin{array}{c|c|c|c}
& \text{Jonas} & ... |
Points $R$, $S$ and $T$ are vertices of an equilateral triangle, and points $X$, $Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices? | 4 | To solve this problem, we need to determine how many noncongruent triangles can be formed using any three of the six points $R$, $S$, $T$, $X$, $Y$, and $Z$, where $X$, $Y$, and $Z$ are the midpoints of the sides of equilateral triangle $\triangle RST$.
#### Step 1: Identify possible triangles
We start by noting that ... |
Find the sum of digits of all the numbers in the sequence $1,2,3,4,\cdots ,10000$. | 180001 |
To find the sum of the digits of all the numbers in the sequence $1, 2, 3, \ldots, 10000$, we can break down the problem into manageable parts based on the number of digits in the numbers.
#### Step 1: Sum of digits for numbers from 1 to 9
Each number from 1 to 9 is a single-digit number. The sum of these digits is s... |
A big $L$ is formed as shown. What is its area? | 22 | 1. **Identify the dimensions of the big rectangle**: The problem states that the shape is a big "L" formed within a rectangle. Assume the dimensions of the big rectangle are $8$ units by $5$ units based on the typical dimensions used in such problems.
2. **Calculate the area of the big rectangle**:
\[
\text{Are... |
Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\overline{AB}$ and $\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$? | 12 | 1. **Identify the Triangle and Hypotenuse**:
Given a right triangle $ABC$ with legs $AB = 20$ and $BC = 21$, we first calculate the length of the hypotenuse $AC$ using the Pythagorean theorem:
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{20^2 + 21^2} = \sqrt{400 + 441} = \sqrt{841} = 29.
\]
2. **Determine the Altit... |
Jerry cuts a wedge from a 6-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters? | 603 | 1. **Identify the dimensions of the cylinder**: The problem states that the cylinder has a radius of $6$ cm. However, the solution incorrectly uses $4$ cm as the radius. We need to correct this and use the correct radius of $6$ cm.
2. **Calculate the volume of the entire cylinder**: The formula for the volume of a cyl... |
In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$ | 17 | 1. **Assume the initial price**: Let's assume the initial price of gasoline at the beginning of January is $P_0 = 100$ dollars.
2. **Price after January's increase**: The price increased by 20% in January. Therefore, the new price at the end of January, $P_1$, is calculated as:
\[
P_1 = P_0 + 0.20 \times P_0 = 1... |
An equilateral triangle and a regular hexagon have equal perimeters. If the triangle's area is 4, what is the area of the hexagon? | 6 | 1. **Assign Variables:**
Let the side length of the equilateral triangle be $s$ and the side length of the regular hexagon be $t$.
2. **Equal Perimeters:**
Since the triangle and the hexagon have equal perimeters, and the triangle has 3 sides while the hexagon has 6 sides, we have:
\[
3s = 6t \implies s = ... |
In racing over a distance $d$ at uniform speed, $A$ can beat $B$ by $20$ yards, $B$ can beat $C$ by $10$ yards, and $A$ can beat $C$ by $28$ yards. Then $d$, in yards, equals: | 100 | 1. **Define Variables:**
Let $a$, $b$, and $c$ be the speeds of $A$, $B$, and $C$ respectively.
2. **Set Up Equations:**
- When $A$ finishes the race, $B$ is 20 yards behind. Thus, in the time $\frac{d}{a}$ it takes for $A$ to finish, $B$ covers $d - 20$ yards. The equation is:
\[
\frac{d}{a} = \frac{d... |
Regular polygons with $5, 6, 7,$ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect? | 68 | To solve this problem, we need to calculate the number of intersection points inside the circle where two sides from different polygons intersect. We consider pairs of polygons and count the intersections for each pair.
1. **Understanding the Intersection Rule**:
- When two polygons with $m$ and $n$ sides ($m > n$)... |
If $\log_{k}{x} \cdot \log_{5}{k} = 3$, then $x$ equals: | 125 | Given the equation:
\[
\log_{k}{x} \cdot \log_{5}{k} = 3
\]
We can use the property of logarithms that states $\log_b a \cdot \log_c b = \log_c a$. Applying this property to our equation, we have:
\[
\log_{k}{x} \cdot \log_{5}{k} = \log_{5}{x}
\]
Thus, the equation simplifies to:
\[
\log_{5}{x} = 3
\]
This implies th... |
In $\triangle ABC$, $AB = 86$, and $AC = 97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is $BC$? | 61 |
#### Step 1: Understanding the Problem
We are given a triangle $\triangle ABC$ with $AB = 86$ and $AC = 97$. A circle centered at $A$ with radius $AB$ intersects line segment $BC$ at points $B$ and $X$. We need to find the length of $BC$, denoted as $BX + CX$, where both $BX$ and $CX$ are integers.
#### Step 2: Apply... |
In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game average for the $10$ ga... | 40 | 1. **Calculate the total points from the first 8 games:**
\[
7 + 4 + 3 + 6 + 8 + 3 + 1 + 5 = 37
\]
2. **Determine the points needed in the ninth game for an integer average:**
- The total points after 9 games must be a multiple of 9 for the average to be an integer.
- The closest multiple of 9 that is g... |
Let $f(x) = 10^{10x}$, $g(x) = \log_{10}\left(\frac{x}{10}\right)$, $h_1(x) = g(f(x))$, and $h_n(x) = h_1(h_{n-1}(x))$ for integers $n \geq 2$. What is the sum of the digits of $h_{2011}(1)$? | 16089 | 1. **Define the functions and calculate $h_1(x)$:**
- Given $f(x) = 10^{10x}$ and $g(x) = \log_{10}\left(\frac{x}{10}\right)$, we can simplify $g(x)$:
\[
g(x) = \log_{10}\left(\frac{x}{10}\right) = \log_{10}(x) - \log_{10}(10) = \log_{10}(x) - 1
\]
- Now, compute $h_1(x)$:
\[
h_1(x) = g(f... |
Set $A$ has $20$ elements, and set $B$ has $15$ elements. What is the smallest possible number of elements in $A \cup B$? | 20 | 1. **Understanding the Problem**: We are given two sets, $A$ and $B$, with $|A| = 20$ and $|B| = 15$. We need to find the smallest possible number of elements in the union of $A$ and $B$, denoted as $A \cup B$.
2. **Using the Principle of Inclusion-Exclusion**: The principle of inclusion-exclusion for two sets $A$ and... |
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24%... | 48 | 1. **Define Variables:**
Let $p$ be the rate at which Paula paints (in house/hours), $h$ be the combined rate of the two helpers (in house/hours), and $L$ be the duration of the lunch break (in hours).
2. **Set Up Equations:**
From the problem, we can establish the following equations based on the work done each... |
Pauline Bunyan can shovel snow at the rate of $20$ cubic yards for the first hour, $19$ cubic yards for the second, $18$ for the third, etc., always shoveling one cubic yard less per hour than the previous hour. If her driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep, then the number o... | 7 | 1. **Calculate the total volume of snow**: The driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep. Therefore, the total volume of snow is:
\[
4 \text{ yards} \times 10 \text{ yards} \times 3 \text{ yards} = 120 \text{ cubic yards}
\]
2. **Calculate the snow removed each hour**: ... |
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice? | 40 | 1. **Determine the amount of juice from pears and oranges:**
- Miki extracts 8 ounces of pear juice from 3 pears. Therefore, the amount of pear juice per pear is:
\[
\frac{8 \text{ ounces}}{3 \text{ pears}} = \frac{8}{3} \text{ ounces per pear}
\]
- Miki extracts 8 ounces of orange juice from 2 ora... |
It takes $5$ seconds for a clock to strike $6$ o'clock beginning at $6:00$ o'clock precisely. If the strikings are uniformly spaced, how long, in seconds, does it take to strike $12$ o'clock? | 11 | 1. **Understanding the Problem:** The problem states that it takes 5 seconds for a clock to strike 6 times at 6 o'clock. We need to find out how long it takes for the clock to strike 12 times.
2. **Analyzing the 6 o'clock strikes:** When the clock strikes 6 times, there are actually 5 intervals between the strikes (th... |
There are 3 numbers A, B, and C, such that $1001C - 2002A = 4004$, and $1001B + 3003A = 5005$. What is the average of A, B, and C? | 3 | 1. **Isolate $C$ and $B$ in terms of $A$:**
- From the equation $1001C - 2002A = 4004$, we can isolate $C$:
\[
1001C = 2002A + 4004 \implies C = 2A + 4
\]
- From the equation $1001B + 3003A = 5005$, we can isolate $B$:
\[
1001B = 5005 - 3003A \implies B = -3A + 5
\]
2. **Calculate t... |
Each of the letters $W$, $X$, $Y$, and $Z$ represents a different integer in the set $\{ 1,2,3,4\}$, but not necessarily in that order. If $\frac{W}{X} - \frac{Y}{Z}=1$, then the sum of $W$ and $Y$ is | 7 | 1. **Identify the possible values for fractions:** Given that $\text{W}, \text{X}, \text{Y}, \text{Z}$ are different integers from the set $\{1, 2, 3, 4\}$, we need to find pairs $(\text{W}, \text{X})$ and $(\text{Y}, \text{Z})$ such that $\frac{\text{W}}{\text{X}} - \frac{\text{Y}}{\text{Z}} = 1$.
2. **List possible ... |
How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11? | 5 | 1. **Formulate the congruences:** We are given that a number $n$ leaves a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11. We can express these conditions using congruences:
\[
n \equiv 2 \pmod{6}, \quad n \equiv 5 \pmod{9}, \quad n \equiv 7 \pmod{11}
... |
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? | 28 | 1. **Label the seats and calculate unrestricted seating for Derek and Eric:**
Label the seats from left to right as $1$ through $5$. Without any restrictions, Derek and Eric can be seated in $5 \times 4 = 20$ ways, since Derek has 5 choices and Eric has 4 remaining choices.
2. **Calculate the restricted seating f... |
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\frac{1}{2}$ foot from the top face. The second cut is $\frac{1}{3}$ foot below the first cut, and the third cut is $\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labe... | 11 | 1. **Understanding the Problem**: We have a cube with a volume of 1 cubic foot, and it is cut into four pieces by three cuts parallel to the top face. The cuts are made at $\frac{1}{2}$ foot, $\frac{1}{3}$ foot, and $\frac{1}{17}$ foot below each preceding cut. The pieces are then rearranged end to end.
2. **Calculati... |
A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils ... | 11 | 1. **Define Variables:**
Let $s$ be the number of students that bought pencils, $c$ be the cost of each pencil in cents, and $n$ be the number of pencils each student bought. We need to find $c$.
2. **Analyze Given Information:**
- Since a majority of the 30 students bought pencils, we have $s > \frac{30}{2} = 1... |
Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must be painted? | 876 | 1. **Calculate the area of the walls in one bedroom**:
Each bedroom has four walls, two pairs of opposite walls. Each pair consists of:
- Two walls of dimensions $12$ feet (length) by $8$ feet (height).
- Two walls of dimensions $10$ feet (width) by $8$ feet (height).
The total area of the walls in one be... |
A mixture of $30$ liters of paint is $25\%$ red tint, $30\%$ yellow tint and $45\%$ water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture? | 40 | 1. **Calculate the amount of yellow tint in the original mixture:**
The original mixture contains $30\%$ yellow tint in $30$ liters. Therefore, the amount of yellow tint is:
\[
0.30 \times 30 = 9 \text{ liters}
\]
2. **Add the additional yellow tint:**
Five liters of yellow tint are added to the origina... |
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 25$ and $(x-4)^2 + 9y^2 = 81?$ | 27 | 1. **Identify the curves and their equations:**
- The first curve is a circle given by \(x^2 + y^2 = 25\).
- The second curve is an ellipse given by \((x-4)^2 + 9y^2 = 81\).
2. **Substitute \(y^2\) from the circle's equation into the ellipse's equation:**
\[
y^2 = 25 - x^2
\]
Substituting into the el... |
Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour? | 7 | To find Kate's overall average speed, we first need to calculate the total distance she traveled and the total time she spent traveling.
1. **Calculate the distance traveled while biking:**
Kate biked at a speed of 16 mph for 30 minutes. Since speed is in miles per hour and time is in minutes, we convert time to ho... |
A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip? | 24 | 1. **Calculate the total distance of the round trip**:
The student drove $120$ miles to his home and $120$ miles back, so the total distance for the round trip is:
\[
120 \text{ miles} + 120 \text{ miles} = 240 \text{ miles}
\]
2. **Calculate the amount of gasoline used for each leg of the trip**:
- On... |
At his usual rate a man rows 15 miles downstream in five hours less time than it takes him to return. If he doubles his usual rate, the time downstream is only one hour less than the time upstream. In miles per hour, the rate of the stream's current is: | 2 | Let $r$ be the man's rowing speed in still water (in miles per hour) and $w$ be the speed of the stream's current (in miles per hour).
1. **Formulate the equations based on the problem statement:**
- The time taken to row 15 miles downstream (where the speed is $r+w$) is 5 hours less than the time taken to row the ... |
Let $x$ be the least real number greater than $1$ such that $\sin(x) = \sin(x^2)$, where the arguments are in degrees. What is $x$ rounded up to the closest integer? | 13 | To solve the problem, we need to find the smallest real number $x > 1$ such that $\sin(x) = \sin(x^2)$. This equation holds true when $x$ and $x^2$ differ by a multiple of $360^\circ$ or when $x^2$ is equivalent to $180^\circ - x + 360^\circ k$ for some integer $k$. We will analyze each choice given in the problem.
##... |
How many positive integers less than $50$ have an odd number of positive integer divisors? | 7 | To solve this problem, we need to determine how many positive integers less than $50$ have an odd number of positive integer divisors.
1. **Understanding the Divisor Counting**: A number has an odd number of divisors if and only if it is a perfect square. This is because divisors generally come in pairs (e.g., for the... |
Define binary operations $\diamondsuit$ and $\heartsuit$ by $a \diamondsuit b = a^{\log_{7}(b)}$ and $a \heartsuit b = a^{\frac{1}{\log_{7}(b)}}$ for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3 \heartsuit 2$ and $a_n = (n \heartsuit (n-1)... | 11 | 1. **Define the operations and initial condition**: We are given two binary operations $\diamondsuit$ and $\heartsuit$ defined by:
\[
a \diamondsuit b = a^{\log_{7}(b)} \quad \text{and} \quad a \heartsuit b = a^{\frac{1}{\log_{7}(b)}}
\]
The sequence $(a_n)$ is defined recursively with $a_3 = 3 \heartsuit 2... |
For every $3^\circ$ rise in temperature, the volume of a certain gas expands by $4$ cubic centimeters. If the volume of the gas is $24$ cubic centimeters when the temperature is $32^\circ$, what was the volume of the gas in cubic centimeters when the temperature was $20^\circ$? | 8 | 1. **Identify the relationship between temperature and volume**:
Given that for every $3^\circ$ rise in temperature, the volume of the gas expands by $4$ cubic centimeters. This implies that for every $3^\circ$ decrease in temperature, the volume of the gas decreases by $4$ cubic centimeters.
2. **Set up the equat... |
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide? | 5 | 1. **Identify the total number of triangles in each half**: Each half of the figure contains 3 red triangles, 5 blue triangles, and 8 white triangles.
2. **Analyze the red triangles**: When the upper half is folded down, 2 pairs of red triangles coincide. This means 4 red triangles (2 from each half) are involved in c... |
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? | 8 | 1. **Identify the prime factors of $12!$ and their exponents:**
We start by finding the exponents of the prime factors of $12!$. For any prime $p$, the exponent of $p$ in $n!$ is given by:
\[
\sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor
\]
where $\left\lfloor x \right\rfloor$ denotes th... |
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $... | 13 | Let $a$ be the number of $2$-point shots, $b$ be the number of $3$-point shots, and $x$ be the number of free throws. We need to find $x$.
Given:
1. The points from two-point shots equal the points from three-point shots, so $2a = 3b$.
2. The number of free throws is one more than the number of two-point shots, so $x ... |
How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$ | 48 | 1. **Start with the given equation:**
\[
\text{lcm}(5!, n) = 5 \cdot \text{gcd}(10!, n)
\]
Here, $5! = 120$ and $10! = 3628800$.
2. **Express $\text{lcm}(5!, n)$ in terms of $n$:**
Since $n$ is a multiple of $5$, we can write $n = 5m$ for some integer $m$. Then, $\text{lcm}(5!, n) = \text{lcm}(120, 5m)$... |
When simplified, $\log{8} \div \log{\frac{1}{8}}$ becomes: | -1 | 1. **Rewrite the fraction inside the logarithm**: We start by recognizing that $\frac{1}{8}$ can be expressed as $8^{-1}$. Therefore, the expression $\log{\frac{1}{8}}$ can be rewritten using the property of logarithms that $\log{a^{-b}} = -b \log{a}$:
\[
\log{\frac{1}{8}} = \log{8^{-1}} = -\log{8}
\]
2. **Si... |
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle? | 200 | 1. **Identify the relationship between the circle and the rectangle**: The circle is inscribed in the rectangle, which means the diameter of the circle is equal to the height (shorter side) of the rectangle.
2. **Calculate the diameter of the circle**: Given that the radius of the circle is $5$, the diameter is:
\[... |
A teacher gave a test to a class in which $10\%$ of the students are juniors and $90\%$ are seniors. The average score on the test was $84.$ The juniors all received the same score, and the average score of the seniors was $83.$ What score did each of the juniors receive on the test? | 93 | 1. **Assume the total number of students**: Let's assume there are $10$ students in the class for simplicity, as $10\%$ of any number of students being juniors will always be $10\%$ of that number. This assumption makes calculations straightforward without loss of generality.
2. **Calculate the number of juniors and s... |
Let $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}.$ The minimum possible value of $x$ has a prime factorization $a^cb^d.$ What is $a + b + c + d?$ | 31 | 1. **Equation Setup**: We start with the equation given in the problem:
\[
7x^5 = 11y^{13}
\]
We aim to find the minimum possible value of $x$ in terms of its prime factors.
2. **Prime Factorization of $x$**: Assume $x$ has a prime factorization of the form $a^cb^d$. Substituting this into the equation, we... |
Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will exactly two friends visit her? | 54 | 1. **Identify the Periodicity**:
Alice, Beatrix, and Claire visit every 3, 4, and 5 days respectively. The least common multiple (LCM) of 3, 4, and 5 is:
\[
\text{LCM}(3, 4, 5) = 60
\]
This means every 60 days, all three friends visit Daphne together.
2. **Divide the 365-day period**:
Since the visi... |
Heather compares the price of a new computer at two different stores. Store $A$ offers $15\%$ off the sticker price followed by a $\$90$ rebate, and store $B$ offers $25\%$ off the same sticker price with no rebate. Heather saves $\$15$ by buying the computer at store $A$ instead of store $B$. What is the sticker price... | 750 |
Let the sticker price of the computer be denoted as \( x \).
**Store A Calculation:**
- Store A offers a 15% discount, so the price after the discount is \( 0.85x \).
- After applying the additional $90 rebate, the final price at store A becomes \( 0.85x - 90 \).
**Store B Calculation:**
- Store B offers a 25% disco... |
Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible? | 15 | To solve this problem, we can use the "stars and bars" theorem, which is a common combinatorial method to determine the number of ways to distribute $n$ identical items into $k$ distinct groups.
In this problem, Pat wants to buy four donuts (identical items) from three types of donuts: glazed, chocolate, and powdered ... |
Values for $A, B, C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the... | 90 | 1. **Understanding the Problem**: We need to find the number of ways to choose values for $A, B, C, D$ from the set $\{1, 2, 3, 4, 5, 6\}$ without replacement such that the curves $y = Ax^2 + B$ and $y = Cx^2 + D$ intersect.
2. **Condition for Intersection**: The curves intersect if there exists an $x$ such that $Ax^2... |
Last year Mr. Jon Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he had left in state taxes. He paid a total of $\textdollar10500$ for both taxes. How many dollars was his inheritance? | 37500 | 1. **Define the variable**: Let $x$ be the total amount of Mr. Jon Q. Public's inheritance.
2. **Calculate federal taxes**: He pays $20\%$ in federal taxes, which is $0.2x$.
3. **Calculate remaining amount after federal taxes**: After paying federal taxes, he has $x - 0.2x = 0.8x$ left.
4. **Calculate state taxes**:... |
For how many positive integers $n \le 1000$ is$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$not divisible by $3$? | 22 | We are tasked with finding how many positive integers \( n \leq 1000 \) make the expression
\[
\left\lfloor \frac{998}{n} \right\rfloor + \left\lfloor \frac{999}{n} \right\rfloor + \left\lfloor \frac{1000}{n} \right\rfloor
\]
not divisible by 3. We start by analyzing the behavior of the floor function in this context.
... |
For how many positive integers $x$ is $\log_{10}(x-40) + \log_{10}(60-x) < 2$? | 18 | 1. **Identify the domain of the function**:
The expression $\log_{10}(x-40) + \log_{10}(60-x)$ is defined only when both $x-40$ and $60-x$ are positive. This implies:
\[
x-40 > 0 \quad \text{and} \quad 60-x > 0
\]
Simplifying these inequalities, we get:
\[
x > 40 \quad \text{and} \quad x < 60
\... |
A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? | 225 | To find the least number of distinct values in the list, we need to consider how the integers are distributed among these values. Given that the mode occurs exactly 10 times, we want to minimize the number of distinct values while ensuring that no other number appears 10 or more times, which would violate the uniquenes... |
A nickel is placed on a table. The number of nickels which can be placed around it, each tangent to it and to two others is: | 6 | To solve this problem, we need to determine how many nickels can be placed around a central nickel such that each is tangent to the central nickel and to its two neighboring nickels.
1. **Understanding the Problem:**
- Each nickel is a circle, and we are asked to place other nickels around one central nickel.
- ... |
What is the largest difference that can be formed by subtracting two numbers chosen from the set $\{ -16,-4,0,2,4,12 \}$? | 28 | To find the largest difference that can be formed by subtracting two numbers chosen from the set $\{ -16,-4,0,2,4,12 \}$, we need to consider the subtraction $a - b$, where $a$ and $b$ are elements of the set. The goal is to maximize this expression.
1. **Maximize $a$:** The maximum value of $a$ is the largest number ... |
What is the largest quotient that can be formed using two numbers chosen from the set $\{ -24, -3, -2, 1, 2, 8 \}$? | 12 | To find the largest quotient formed using two numbers from the set $\{-24, -3, -2, 1, 2, 8\}$, we need to consider the absolute values of the numbers and the signs to maximize the quotient $\frac{a}{b}$.
1. **Maximizing the Quotient**:
- The quotient $\frac{a}{b}$ is maximized when $a$ is maximized and $b$ is minim... |
The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center? | 130 | 1. **Identify the distance between the third and tenth exits**:
The third exit is at milepost 40 and the tenth exit is at milepost 160. Therefore, the distance between these two exits is calculated as:
\[
160 - 40 = 120 \text{ miles}
\]
2. **Calculate the location of the service center**:
The service c... |
Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$. What is the value of
\[x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?\] | 440 |
Given the equations:
1. \(x + y = 4\)
2. \(x \cdot y = -2\)
We need to find the value of:
\[x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y\]
#### Step-by-step calculation:
1. **Expression Simplification**:
\[x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y = x + y + \frac{x^3}{y^2} + \frac{y^3}{x^2}\]
2. **Using the given ... |
If $3^p + 3^4 = 90$, $2^r + 44 = 76$, and $5^3 + 6^s = 1421$, what is the product of $p$, $r$, and $s$? | 40 | We are given three equations involving exponents and asked to find the product of $p$, $r$, and $s$.
1. **Solving for $p$:**
\[
3^p + 3^4 = 90
\]
We know that $3^4 = 81$. Substituting this into the equation, we get:
\[
3^p + 81 = 90
\]
Subtracting 81 from both sides:
\[
3^p = 9
\]
S... |
On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz? | 3 | 1. **Calculate the percentage of students scoring 100 points**:
Given that $10\%$ scored $70$ points, $35\%$ scored $80$ points, and $30\%$ scored $90$ points, the percentage of students scoring $100$ points is:
\[
100\% - (10\% + 35\% + 30\%) = 100\% - 75\% = 25\%
\]
2. **Determine the median score**:
... |
Samuel's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exact... | 20 | To solve this problem, we need to determine how many of the smaller $1 \times 1 \times 1$ inch cubes have icing on exactly two sides. We will analyze the positions of these cubes on the cake.
1. **Understanding the Cake Structure**:
- The cake is a $4 \times 4 \times 4$ cube.
- Icing is on the top and all four v... |
If for any three distinct numbers $a$, $b$, and $c$ we define $f(a,b,c)=\frac{c+a}{c-b}$, then $f(1,-2,-3)$ is | 2 | 1. We are given the function \( f(a, b, c) = \frac{c+a}{c-b} \) and need to evaluate \( f(1, -2, -3) \).
2. Substitute \( a = 1 \), \( b = -2 \), and \( c = -3 \) into the function:
\[
f(1, -2, -3) = \frac{-3 + 1}{-3 - (-2)}.
\]
3. Simplify the numerator and the denominator:
\[
\frac{-3 + 1}{-3 + 2} = ... |
When the mean, median, and mode of the list
\[10,2,5,2,4,2,x\]
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$? | 20 | 1. **Calculate the Mean**:
The mean of the list $10, 2, 5, 2, 4, 2, x$ is calculated as follows:
\[
\text{Mean} = \frac{10 + 2 + 5 + 2 + 4 + 2 + x}{7} = \frac{25 + x}{7}.
\]
2. **Determine the Mode**:
The mode is the number that appears most frequently in the list. Here, the number $2$ appears three t... |
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise.
In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise... | 6 | 1. **Identify the movement of Alice and Bob**:
- Alice moves 5 points clockwise.
- Bob moves 9 points counterclockwise, which is equivalent to moving 3 points clockwise (since 9 counterclockwise on a 12-point circle is the same as 3 clockwise: $12 - 9 = 3$).
2. **Calculate the effective movement per turn**:
... |
How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ so that the mean (average) of the remaining numbers is 6? | 5 |
1. **Calculate the total sum of the original set**:
The set given is $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$. The sum of all elements in this set can be calculated using the formula for the sum of an arithmetic series:
\[
\text{Sum} = \frac{n}{2} \times (\text{first term} + \text{last term})
\]
where $... |
If the side of one square is the diagonal of a second square, what is the ratio of the area of the first square to the area of the second? | 2 | 1. **Assign Variables:**
Let the side length of the second square be $s_2$. Then, the diagonal of this square, which is also the side length of the first square, is denoted as $s_1$.
2. **Relate Diagonal to Side Length:**
The diagonal of a square can be calculated using the Pythagorean theorem. For the second sq... |
The sum of seven integers is $-1$. What is the maximum number of the seven integers that can be larger than $13$? | 6 | To solve this problem, we need to determine the maximum number of integers among the seven that can exceed 13 while still achieving a total sum of $-1$.
1. **Assume the maximum number of integers greater than 13**: Let's denote these integers that are greater than 13 as $x_1, x_2, \ldots, x_k$ where $k$ is the number ... |
Part of an \(n\)-pointed regular star is shown. It is a simple closed polygon in which all \(2n\) edges are congruent, angles \(A_1,A_2,\cdots,A_n\) are congruent, and angles \(B_1,B_2,\cdots,B_n\) are congruent. If the acute angle at \(A_1\) is \(10^\circ\) less than the acute angle at \(B_1\), then \(n=\) | 36 | 1. **Understanding the Star Polygon**: In the given problem, we have a regular star polygon with $n$ points. Each point of the star has two angles associated with it: one at $A_i$ and one at $B_i$ for $i = 1, 2, \ldots, n$. All $A_i$ angles are congruent, and all $B_i$ angles are congruent. The acute angle at each $A_i... |
The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? | 7 | 1. **Identify the relationship between the angles**: In a right triangle, the sum of the angles is $180^\circ$. Given one angle is $90^\circ$, the sum of the other two angles, $a^\circ$ and $b^\circ$, must be $90^\circ$. Thus, we have:
\[
a^\circ + b^\circ = 90^\circ
\]
2. **Condition on $a$ and $b$**: Both $... |
Connie multiplies a number by 2 and gets 60 as her answer. However, she should have divided the number by 2 to get the correct answer. What is the correct answer? | 15 | Let's denote the number Connie should have used as $x$. According to the problem, Connie mistakenly multiplied $x$ by $2$ instead of dividing it by $2$. This gives us two equations based on her actions and what she should have done:
1. **Mistaken Calculation:**
\[ 2x = 60 \]
2. **Correct Calculation:**
\[ \frac... |
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages? | 18 | 1. **Define the ages**: Let Jack's age be represented as $\overline{ab} = 10a + b$ where $a$ and $b$ are the tens and units digits respectively. Similarly, Bill's age is $\overline{ba} = 10b + a$.
2. **Future ages**: In five years, Jack's age will be $10a + b + 5$ and Bill's age will be $10b + a + 5$.
3. **Given cond... |
Each vertex of a cube is to be labeled with an integer 1 through 8, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How ma... | 6 | To solve this problem, we need to label each vertex of a cube with integers from $1$ to $8$ such that the sum of the numbers on the vertices of each face is the same. Additionally, we consider two arrangements the same if one can be obtained from the other by rotating the cube.
#### Step 1: Calculate the total sum and... |
Find the sum of the arithmetic series
\[20+20\frac{1}{5}+20\frac{2}{5}+\cdots+40\] | 3030 | 1. **Identify the series type and formula**: The given series is an arithmetic series. The sum of an arithmetic series where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms can be calculated using the formula:
\[
S = \frac{n(a_1 + a_n)}{2}
\]
2. **Prove the formula for the sum ... |
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is | 175 | 1. **Identify the given information**: We are given two congruent squares $ABCD$ and $EFGH$ with side length $AB = 10$. The point $G$ is the center of square $ABCD$.
2. **Calculate the area of each square**:
- The area of square $ABCD$ is $AB^2 = 10^2 = 100$.
- Since $EFGH$ is congruent to $ABCD$, its area is a... |
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number? | 8 | 1. Let $a$ be the integer written down two times, and $b$ be the integer written down three times. According to the problem, the sum of these five numbers is 100. Therefore, we can write the equation:
\[
2a + 3b = 100
\]
2. We know that one of the numbers, either $a$ or $b$, is 28. We first assume $a = 28$ an... |
Let $a$, $b$, $c$, $d$, and $e$ be distinct integers such that
$(6-a)(6-b)(6-c)(6-d)(6-e)=45$
What is $a+b+c+d+e$? | 25 | 1. **Identify the factors of 45**: We start by noting that the equation $(6-a)(6-b)(6-c)(6-d)(6-e)=45$ implies that the expressions $(6-a), (6-b), (6-c), (6-d), (6-e)$ are integers whose product is 45. We need to find distinct integers $a, b, c, d, e$ such that this condition is satisfied.
2. **Factorize 45**: The int... |
Portia's high school has $3$ times as many students as Lara's high school. The two high schools have a total of $2600$ students. How many students does Portia's high school have? | 1950 | 1. **Define Variables:**
Let $P$ represent the number of students in Portia's high school and $L$ represent the number of students in Lara's high school.
2. **Set Up Equations:**
From the problem, we know:
\[ P = 3L \]
\[ P + L = 2600 \]
3. **Substitute and Solve for $L$:**
Substitute $P = 3L$ into the... |
Monica is tiling the floor of her 12-foot by 16-foot living room. She plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will she use? | 87 | 1. **Calculate the number of one-foot by one-foot tiles for the border:**
- The border consists of tiles around the perimeter of the room. Since the border is one tile wide, the length of the room minus two tiles (one for each end) will be covered by the border on each side.
- The length of the room is 16 feet, s... |
The sum of the numerical coefficients in the complete expansion of $(x^2 - 2xy + y^2)^7$ is: | 0 | 1. **Identify the Expression**: The given expression is $(x^2 - 2xy + y^2)^7$.
2. **Recognize the Binomial Expansion**: Observe that $x^2 - 2xy + y^2$ can be rewritten as $(x-y)^2$. Therefore, the expression becomes $((x-y)^2)^7 = (x-y)^{14}$.
3. **Apply the Binomial Theorem**: The binomial theorem states that $(a+b... |
Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$? | 30 | 1. **Define the sets and probabilities**: Let $A$ be the set of all configurations where the balls are distributed as $3{-}5{-}4{-}4{-}4$ among the bins, and let $B$ be the set of all configurations where the balls are distributed as $4{-}4{-}4{-}4{-}4$. Define $p = \frac{|A|}{N}$ and $q = \frac{|B|}{N}$, where $N$ is ... |
How many ordered pairs of integers $(x, y)$ satisfy the equation $x^{2020} + y^2 = 2y$? | 4 |
1. **Rearrange and Complete the Square**:
Start by rearranging the given equation:
\[
x^{2020} + y^2 = 2y.
\]
We can rewrite this as:
\[
x^{2020} + y^2 - 2y = 0.
\]
Completing the square for \(y\), we have:
\[
x^{2020} + (y-1)^2 - 1 = 0 \implies x^{2020} + (y-1)^2 = 1.
\]
2. **Anal... |
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